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Quadruple fixed point theorems for nonlinear contractions in partially ordered metric spaces. (English) Zbl 1264.47057
Let $$(X,d,\leq)$$ be a complete ordered metric space, $$g:X\to X$$ and $$F:X^4\to X$$ have the mixed $$g$$-monotone property (i.e., $$F$$ is $$g$$-nondecreasing in the first and the third and $$g$$-nonincreasing in the second and the fourth variables) and satisfy the condition $d(F(x,y,z,w),F(u,v,r,t))\leq\phi(\frac14(d(gx,gu)+d(gy,gv)+d(gz,gr)+d(gw,gt)))$ whenever $$gx\leq gu$$, $$gy\geq gv$$, $$gz\leq gr$$ and $$gw\geq gt$$, where $$\phi:[0,\infty)\to[0,\infty)$$ satisfies $$\phi(t)<t$$ for $$t>0$$ and $$\lim_{r\to t+}\phi(r)<t$$ for each $$t>0$$. Under some additional conditions, the authors prove that there exist $$x,y,z,w\in X$$ such that $F(x,y,z,w)=gx,\quad F(x,w,z,y)=gy, \quad F(z,y,x,w)=gz, \quad F(z,w,x,y)=gw,$ i.e., $$F$$ and $$g$$ have a so-called quadruple coincidence point. The uniqueness of such a point is also obtained under certain conditions. Some very easy examples are presented.

##### MSC:
 47H10 Fixed-point theorems 46J10 Banach algebras of continuous functions, function algebras 54H25 Fixed-point and coincidence theorems (topological aspects)
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